I want to try my hand at designing quantum algorithms to solve certain problems. I feel like I understand (for example) how Grover's algorithm and Shor's algorithm work, and I'm excited to apply the various "computing tricks" that aren't available in classical computing. Unfortunately, my knowledge of such tricks is limited to their application in the few famous algorithms, and I'm not at all sure how to legitimately apply them in general.

Are there any resources available that will quickly bring me up to speed?

4 Answers
4

In the 1990's it was realized that
quantum physics has some spectacular
applications in computer science. This
book is a concise introduction to
quantum computation, developing the
basic elements of this new branch of
computational theory without assuming
any background in physics. It begins
with an introduction to the quantum
theory from a computer-science
perspective. It illustrates the
quantum-computational approach with
several elementary examples of quantum
speed-up, before moving to the major
applications: Shor's factoring
algorithm, Grover's search algorithm,
and quantum error correction. The book
is intended primarily for computer
scientists who know nothing about
quantum theory, but will also be of
interest to physicists who want to
learn the theory of quantum
computation, and philosophers of
science interested in quantum
foundational issues.

Since you understand Grover's algorithm and Shor's algorithm, you're a lot closer to being up to speed than you might guess. Specifically, the field of quantum algorithms is fairly narrow, much more so than classical algorithms, so getting caught up is not as daunting. Aside from generalities like using quantum computers to simulate quantum systems, there are only three or four main types of quantum algorithms (at a broad level like "quantum Fourier sampling").

The right place to start is books like Mermin or Nielsen and Chuang, as in the other answers, but after that you'll have to move to papers. Stephen Jordan has compiled a pretty comprehensive list of about two hundred quantum algorithms papers in the Quantum Algorithms Zoo, organized by problem. Browsing through this list is a good way to get an overview of what's out there and choose some papers relevant to your interests.

Also, the course by Prof. $Vazirani$ on $edx$ "(www.edx.org)" is good way for learning the necessary material in this way.

Since, there are two main algorithms and other algorithms uses these basic algorithms, you can understand deeply these two quantum algorithms and modify your problem again to find an approach. These two basic and important algorithms are Grover's algorithm (for searching problem) and Shor's algorithm (for factorization). Also, Simons algorithm is good for studying. But, in general, you must learn Fourier sampling and superposition and the quantum gates and their works.